I took my prelim analysis about 3 years ago and I remember wanting to write down all of my solutions so that I could share it later for anyone else who is studying for prelims in the future, but ultimately I didn’t do any of that 😛

During my studies I decided that part of the prelims is to

1. Teach you how to be resourceful i. e. How to use Math Stack Exchange, search Math Overflow, or maybe look for analysis practice exams on other University websites

and

2. I think it’s important to get comfortable with the feeling of being stuck on a problem for a few days – because this is what it’s like in the real research world.

And so I didn’t write down any of my Solutions ( partially I was kinda lazy too – please don’t judge me! haha ). However, I did want to leave some links to a few of the resources that I used whenever I was.

- Math Stack Exchange
- https://math.stackexchange.com/questions/216858/positive-operator-is-bounded
- https://math.stackexchange.com/questions/803293/show-that-t-is-continuous-with-langle-x-tx-rangle-geq-0?lq=1
- https://math.stackexchange.com/questions/156609/continuous-projections-to-finite-dimensional-subspaces-of-normed-spaces
- https://math.stackexchange.com/questions/1678615/why-is-a-a-compact-operator
- https://math.stackexchange.com/questions/265348/do-weak-convergence-and-convergence-of-norms-imply-convergence-in-l2
- https://math.stackexchange.com/questions/758462/a-is-a-hermitian-projection-if-and-only-if-it-is-an-orthogonal-projection

- Math Overflow
- Proof-Wiki
- Other Math Dept. pages’ prelim resources
- UC Berkeley
- UC Davis
- UW – Madison (Prof. Seeger’s notes, but the dept. page also has useful information)

- Textbooks
- I used Folland
- (but you can find your favorite textbooks online – for example at libgen https://sites.google.com/site/themetalibrary/library-genesis )

- Also, asking other grad students!
- I can’t overstate how much help I was kindly given from the more senior graduate students.